In recent decades, there has a great interest on the connection between kinetic equations and their fluid limits in different scalings.
In the late 1980s, the so-called BGL program was initialized by Bardos-Golse-Levermore [1]}, [2]}, which aimed at obtaining rigorously the Leray's solutions to incompressible Navier-Stokes equations from the Boltzmann equation with large initial data in the framework of DiPerna-Lions renormalized solutions [3]}. This program was finally completed by Golse and Saint-Raymond [4]}, [5]}. For incompressible Euler limits, Saint-Raymond made the main contributions in [6]}, [7]}, which connected the DiPern-Lions solutions to the dissipative solutions. As for the hyperbolic scaling, say, acoustic limit, Golse-Levermore [8]} established the acoustic limit with an assumption on the size of the fluctuations. Later on, the restriction has been relaxed to the borderline in [9]}.
